The influence of floating ice on the dynamic behavior of ships and offshore structures depends on many factors such as ice thickness and its relative speed with respect to the floating structure. The ice resistance to ship motion forms an essential problem in ship design and navigation. Furthermore, local or global ice loads acting on ocean systems are random and nonsmooth when impact interaction takes place. Impact loads on the bow of a ship navigating in solid ice may be modeled by a Poisson law. The measured stress amplitudes on the ship frame at the bow follow an exponential distribution. The nonhomogeneity and difference in ice microstructure, as well as the influence of salt and temperature, result in a great uncertainty in the ice strength. Therefore, the current review article aims at assessing the ice related problems encountered by offshore structures as well as by ships during their navigation. It also discusses the impacts of local and global ice loads on floating structures and reviews their existing probabilistic models. Moreover, this article covers the dynamic interaction of ice with flexible and rigid structures, and ships. In view of ice loads on marine systems, new design regulations have been introduced by international organizations that are involved in the design and building of ships as well as offshore structures. The ship stochastic stability and the first-passage roll stabilization problem associated with random ocean waves will also be described in an attempt to stimulate future research work dealing with ice impact loads. Moreover, due to the lack of research activities addressing the control problem of ships operating in icy waters, the current article will briefly discuss passive and active control schemes developed for controlling the ship roll motion. There are 529 references cited in this review article.